Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

TL;DR

This paper extends Weyl quantization to degenerate pre-symplectic forms and enlarged Poisson algebras, constructing a Banach-* algebra framework that leads to strict deformation quantization in the $C^*$-algebraic sense.

Contribution

It introduces a novel approach to deformation quantization using measure-based Banach-* algebras on infinite-dimensional spaces, generalizing existing $C^*$-algebraic methods.

Findings

Achieved strict deformation quantization within Banach-* algebras.

Connected measure Banach-* algebras with $C^*$-algebraic quantization.

Demonstrated functorial dependence of Weyl quantization on test function spaces.

Abstract

$C^*$-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-$*$-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-$*$-algebras instead of $C^*$-algebras. Fourier transformation and representation theory of the measure Banach-$*$-algebras are combined with the theory of continuous projective group representations to arrive at the genuine $C^*$-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter $\hbar$. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.